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Pentagon relation and Biedenharn-Elliott identity

Philipp Korablev

Abstract

The main subject of the paper is the pentagon relation. This relation can be expressed in different ways. We start with the natural geometric form of the pentagon relation. Then we express it in algebraic form as a family of equations with a set of linear maps as variables. Next, we derive several equivalent forms of the algebraic pentagon relation. These forms can be expressed using the classical notion of 6j-symbols. Finally, we show how to extract a solution of the pentagon relation from any modular category.

Pentagon relation and Biedenharn-Elliott identity

Abstract

The main subject of the paper is the pentagon relation. This relation can be expressed in different ways. We start with the natural geometric form of the pentagon relation. Then we express it in algebraic form as a family of equations with a set of linear maps as variables. Next, we derive several equivalent forms of the algebraic pentagon relation. These forms can be expressed using the classical notion of 6j-symbols. Finally, we show how to extract a solution of the pentagon relation from any modular category.
Paper Structure (7 sections, 3 theorems, 48 equations, 5 figures)

This paper contains 7 sections, 3 theorems, 48 equations, 5 figures.

Table of Contents

  1. Geometric form of the pentagon relation
  2. Algebraic form of the pentagon relation
  3. Three forms of the pentagon relation
  4. Component form
  5. Tensor form
  6. Normalized tensor form
  7. Solution of the pentagon relation from modular categories

Key Result

Theorem 1

Let the family $\{\mathcal{F}_{abc}^{d}\}$ be a solution of the pentagon relation. Then for each $a, b, c, d, e, x, y, p, q\in I$ the components $\{\mathcal{F}_{abc}^{d}|^{x}_{y}\}$ satisfy to the following equation:

Figures (5)

  • Figure 1: Geometric form of the pentagon relation
  • Figure 2: Coloured oriented triangle corresponds to the module $V_{ab}^c$
  • Figure 3: The coloured square corresponds to the tensor product $V_{xc}^d\otimes V_{ab}^x$ (on the left) and to the tensor product $V_{ay}^d\otimes V_{bc}^y$ (on the right).
  • Figure 4: The square on the left corresponds to the module $\bigoplus\limits_{x\in I}V_{xc}^d\otimes V_{ab}^x$, and the square on the right corresponds to the module $\bigoplus\limits_{y\in I}V_{ay}^d\otimes V_{bc}^y$
  • Figure 5: Different triangulations of the coloured pentagon and corresponding modules

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof